differentiate x^3y three times with respect to x

1) Let's differentiate:

\frac{d}{d x} (x^{3} y)

ⓘ For differentiation with respect to

x

y

is treated as constant. A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (x^{3} y)

y \cdot \frac{d}{d x} (x^{3})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{3})

3 x^{2}

y \cdot 3 x^{2}

2) Let's perform the differentiation:

\frac{d}{d x} (3 x^{2} y)

ⓘ For differentiation with respect to

x

3 y

is treated as constant. A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (3 x^{2} y)

3 y \cdot \frac{d}{d x} (x^{2})

3 y \cdot \frac{d}{d x} (x^{2})

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

3 y \cdot 2 x

ⅰ

3 \cdot 2

6

6 y x

3) Let's perform the differentiation:

\frac{d}{d x} (6 x y)

ⓘ For differentiation with respect to

x

6 y

is treated as constant. A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (6 x y)

6 y \cdot \frac{d}{d x} (x)

6 y \cdot \frac{d}{d x} (x)

ⓘ The derivative of x is 1:

\frac{d}{d x} (x)

1

6 y \cdot 1

▶Derivative:

6 y